I am always amazed by the** idea of being at the minimum. **Because every physical system near the minimum is an **harmonic oscillator**, and we love harmonic oscillator !

Let us consider a generic **potential energy function** and suppose that is a **stationary point** for . A stationary point could be a minimum, maximum or inflection point of a differentiable function, in which the first derivative is null. In other words in a stationary point the function stops to increase or decrease.

If we expand in Taylor series the potential in to the second order we have:

Since is a stationary point for the first derivative is null and the potential becomes:

The first term is a number equal to the value of the potential in the stationaty point. Since a generic potential energy is defined up to an additive constant, we can put . Furthermore, the expression means if we stop the expansion of the potential to the second order we commit an error that is an infinitesima quantity lower than .

Finally, we can express the potential energy near a stationary point in this form:

(where ) that is the potential energy of a **harmonic oscillator** !

The harmonic oscillator is wherever you look in theoretical Physics (Quantum Mechanics, Quantum Field Theory , Solid State Physics and so on …).

The most important thing is we are able to **exactly** solve the harmonic oscillator, and then you can solve every physical system near a stationary point.

Lennard – Jones Energy Potential (source: Wikipedia)

For example, consider the **Lennard – Jones potential**, phenomenologically describing the short range interaction between neutral particles:

where is the depth of the potential, is the particles relative distance and is the stationary point. If we expand the LJ potential near the minimum we have:

(where )

When the inter-particle distance is approximately equal to they vibrate in harmony !

So, if you stay in the minimum of your potential energy, *sin prisa, *you can vibrate *sin pausa (cit. R.B)*

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